The mathematical community often considers a very short, highly focused paper to be a note. A few journals officially categorize the papers they publish into notes and full papers, but usually no formal distinction is made. There is no clear dividing line between these cases in general. Typically it comes down to the author's intent: a note is just meant to record some observation in the literature, even though it may not be important enough to justify a lengthy or detailed paper, while anything else is a full paper. Nash's paper would probably not be considered a note, even though it's short and focused, because it's too important (and arguably wouldn't have benefited from being longer). The Watson and Crick paper is not so relevant, since papers from other scientific fields have completely different length distributions from mathematics papers.
I'm not fond of the term "true research paper" in the referee report, since it suggests that there's a clearer distinction between notes and papers. But what the referee means is simple: you have found something worth recording as a note, but not yet enough to justify the length of your paper (in the referee's opinion).
Lastly, how does publishing a note vs a research paper affect a scholar, e.g. prestige and/or income?
The prestige depends on the note/paper. The ceiling is vastly higher for papers: the world's best papers are far more impressive than the world's best notes. If you are looking for a job at a major research university, then publishing a note might help a little, but ultimately you will need to have really good papers. On the other hand, there are plenty of mediocre papers out there, and a good note can be better than a not so good paper.
As for income, the relationship between publishing and income is tricky. To the extent your publications help get you a job, promotion, or raise, they can make a real difference for your income. Aside from that, you won't make any money from them. In any case, it's difficult to quantify their effect on income.